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Formulate the following problem as least squares problems. For each problem, give a matrixA and a vector b such that the problem can be expressed asargmin)‖Ax − b‖..(you are not asked to solve the problems. Just state define matrix A and vector b)a. minimize x/. + 2x.. + 3x3. + (x/ − x. + x3 − 1). + (−x/ − 4x. + 2).;b. minimize x8x + ||Bx − d||., where the p × n matrix B and the p-vector d are givenc. minimize ||Bx − d||.+2||Fx − g||.. The p × n matrix B, the ???? × n matrix F, the p-vector d and the ????-vector g are given.

Respuesta :

Answer:

a) [tex]A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right][/tex]

[tex]b=\left[\begin{array}{ccc}0\\1\end{array}\right][/tex]

b) [tex]||Ax-b||^{2} =(-bx_{2}+4)^{2} (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}[/tex]

c) [tex]A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right][/tex]

[tex]x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right][/tex]

[tex]b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right][/tex]

Step-by-step explanation:

a) considering the equation:

Minimize [tex]x_{1}^{2} +2x_{2}x^{2} +3x_{3}^{2}+(x_{1} -x_{2} +x_{3} -1)^{2} +(-x_{1} -4x_{2} +2)^{2}[/tex]

[tex]A=\left[\begin{array}{ccc}1&2&3\\1&-1&1\end{array}\right][/tex] (matrix A)

vector b

[tex]b=\left[\begin{array}{ccc}0\\1\end{array}\right][/tex]

b) If Pxn is matrix B and p-vector d, we have:

minimize [tex](-6x_{2}+4)^{2} +(-4x_{1} +3x_{2} -1)+(x_{1} +8x_{2} -3)^{2}[/tex]

[tex]Ax=\left[\begin{array}{ccc}0&-6&0\\-4&3&0\\1&8&0\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right][/tex]

[tex]b=\left[\begin{array}{ccc}-4\\1\\3\end{array}\right][/tex]

[tex]Ax-b=\left[\begin{array}{ccc}-bx_{2}+4 \\-4x_{1}+3x_{2}-1 \\x_{1}+8x_{2}-3 \end{array}\right] =1[/tex]

[tex]||Ax-b||^{2} =(-bx_{2}+4)^{2} (-4x_{1} +3x_{2} -1)^{2} +(x_{1} +8x_{2} -3)^{2}[/tex]

c) minimize [tex]2(-bx_{2}+4)^{2} +3(-4x_{1} +3x_{2} -1)^{2} +4(x_{1} -x_{2} -3)^{2} -(6\sqrt{2}x_{2} +4\sqrt{2} )^{2} +(-4\sqrt{3} x_{1} +3\sqrt{3}x_{2} -\sqrt{3})^{2} +(2x_{1} -16x_{2} -6)^{2}[/tex]

in matrix:

[tex]A=\left[\begin{array}{ccc}0&6\sqrt{2} &0\\\sqrt{3} &3\sqrt{3} &0\\2&-16&0\end{array}\right][/tex]

[tex]x=\left[\begin{array}{ccc}x_{1} \\x_{2} \\x_{3} \end{array}\right][/tex]

[tex]b=\left[\begin{array}{ccc}-\sqrt{2} \\\sqrt{3} \\6\end{array}\right][/tex]

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